The Propagation And Transformation Of Continue Optical Waves And Optical Pulses On Curved Surface

The properties of gravity and curved space-time can be completely described by general relativity.Astronomical observations of massive celestial bodies are a proof of general relativity.In addition,another popular research direction is to study the two-dimensional surface as a submanifold embedded in a high-dimensional manifold.The Maxwell equations in the dielectric and the covariant Maxwell equations in the curved space have a strong similarity in form.In recent years,researchers have done a lot of work on optical transmission on general manifolds,and the potential and applications of this concept in the field of optical communications have gradually begun to be discovered.Borrowing some mathematical formulas of the same form,by confining light on a two-dimensional surface,complex optical problems in curved space can be simplified to a large extent.This article starts from the classic perfect imaging device Maxwell fisheye,and derives the current general methods for the research of curved surface light transmission.Subsequently,the description of curved surfaces was extended to Riemannian geometry,and concepts such as metric tensor,Rich tensor,scalar curvature,and extrinsic curvature used to describe curved surfaces were introduced.Based on the geometric language description of the curved surface and using the geometric optics approximation,we derive the eikonal expression and the point source propagation function of light propagating on a rotationally symmetric surface with constant Gaussian curvature.The results show that the shape of the curved surface determines the light propagation behavior.The curved surface of positive Gaussian curvature can act as a focusing lens,and under certain conditions,it can offset the divergence of Gaussian light to obtain a stable transmission beam with the same spot radius.For rotationally symmetric surfaces,the periodicity of light and the set of eigenfrequency are involved.The calculation results show that when the radius of curvature of the Gaussian curvature is inconsistent with the radius of curvature of the equator,the light on the surface has only a unique eigenfrequency;for a spherical surface,the radius of curvature of the Gaussian curvature is equal to the radius of curvature of the equator,then any type of initial beam spot can exist stably on the curved surface.At the same time,we discussed the consistency between the optical transmission on the curved surface and the fractional Fourier transform.Further,we studied the transmission of ultrashort pulse lasers on curved surfaces.Using the Green's function of optical transmission on the curved surface,the transmission characteristics of the ultrashort pulse with Gaussian distribution on the curved surface are derived.After giving an intuitive schematic diagram,the evolution of some typical physical quantities of the pulse along with the propagation process is analyzed in detail.Then,starting from the vector inner product relationship of the surface,the relationship that needs to be satisfied by the group velocity control in the surface space is deduced.The transmission constant pulse on the curved surface is given,and it is found that the retardation potential of the ultrashort pulse on the curved surface is no longer proportional to the transmission distance.At the same time,the group velocity control characteristics of the invariant pulses transmitted on the curved surface are described.We hope that our theoretical research will help people understand the concept of curved surface and curved surface light transmission,and provide a reference for the production of two-dimensional micro-integrated optical paths.